GRACO 05
Searching for geodetic boundary vertices sets
Ignacio M Pelayo
Departament de Matemàtica Aplicada 3
Universitat Politècnica de Catalunya
Barcelona, Spain
Joint work with:
Carmen Hernando, Mercè Mora, Carlos Seara
Jose Cáceres, M. Luz Puertas
2
contour
eccentricity
extreme set
boundary
periphery
3
c
a
b
d
3
Ext(G)={a}
4
f
j
4
5
e
i
h
g
3
4
5
3
3
4
Per(G)={a,f}
Ecc(G)={a,c,d,e,f,g,h}
Ct(G)={a,c,f}
Bd(G)=V-b
4
zz
5
zz
6
[abcd]=[periphery,contour,eccentricity,boundary]
7
(ecc(1), ecc(2), ecc(3), ecc(4), ecc(5), ecc(6))=(3,2,2,3,3,2)
8
9
10
11
12
13
14
CONVEX SET RECOVERING PROCEDURE
15
16
CHORDAL GRAPH
No induced cycles of length greater than 3
Intersection graph: Subtrees of a tree
Our main result:
Every g-convex set is the geodetic closure of its contour
In particular:
If G=(V,E) is chordal, then I[Ct(G)]=V
CHORDAL GRAPHS ARE PERFECT
What about other perfect families?
c
a
b
d
e
f
j
Ct(G)={a,c,f}
I[Ct(G)]=V-j
i
h
g
G is not perfect
G+id is perfect (comparability)
G+id+eh is perfect (permutation)
What about the bipartite family?
AT-free
co-comp.
perfect
comparability
parity
bipartite
trapezoid
cochordal
chordal
D.H.
str. chordal
catval
permutation
undirected path
directed path
cograph
ptolemaic
circular arc
split
tree
interval
GEODETIC CONTOURS
complete
19
GRACO 05
Searching for geodetic boundary vertices sets
Ignacio M Pelayo, Carmen Hernando, Mercè Mora,
Carlos Seara Jose Cáceres, M. Luz Puertas
Obrigado Thanks
Gracias